Remarks on subcategories of artinian modules

TL;DR

This paper investigates the structure of subcategories within artinian modules, establishing that all wide subcategories are Serre subcategories and describing a bijection with certain closed subsets of prime ideals.

Contribution

It proves that every wide subcategory of artinian modules is a Serre subcategory and characterizes Serre subcategories via specialization closed subsets of prime ideals.

Findings

All wide subcategories are Serre subcategories.

A bijection exists between Serre subcategories and specialization closed subsets.

The results connect subcategory structures with prime ideal topology.

Abstract

We study two subcategories of the category of artinian modules, a wide subcategory and a Serre subcategory. We prove that all wide subcategories of artinian modules are Serre subcategories. We also provide the bijection between the set of Serre subcategories and the set of specialization closed subsets of the set of closed prime ideals of some completed ring.